Question: An amusement park ride consists of a vertical cylinder that
spins about a vertical axis. When the cylinder spins sufficiently fast, any person
inside it is held up against the wall. Suppose that the coefficient of static
friction between a typical person and the wall is . Let the mass of an typical
person be
, and let be the radius of the cylinder.
Find the critical angular velocity of the cylinder above which a typical person
will not slide down the wall. How many revolutions per second
is the cylinder executing at this critical velocity?

Answer: In the vertical direction, the person is subject to
a downward force due to gravity, and a maximum upward force due
to friction with the wall. Here, is the normal reaction between the person
and the wall. In order for the person not to slide down the wall, we require
. Hence, the critical case corresponds to

In the radial direction, the person is subject to a single force: namely, the reaction
due to the wall, which acts radially inwards. If the cylinder (and, hence, the person)
rotates with angular velocity , then this force must provided the acceleration
towards the axis of rotation. Hence,